A165998 -id:A165998 - cp's OEIS frontend

A074377 Generalized 10-gonal numbers: m(4m - 3) for m = 0, +- 1, +- 2, +- 3, ...

0, 1, 7, 10, 22, 27, 45, 52, 76, 85, 115, 126, 162, 175, 217, 232, 280, 297, 351, 370, 430, 451, 517, 540, 612, 637, 715, 742, 826, 855, 945, 976, 1072, 1105, 1207, 1242, 1350, 1387, 1501, 1540, 1660, 1701, 1827, 1870, 2002, 2047, 2185, 2232, 2376, 2425

Offset: 0

W. Neville Holmes, Sep 04 2002

Also called generalized decagonal numbers.
Odd triangular numbers decremented and halved.
It appears that this is zero together with the partial sums of A165998. - Omar E. Pol, Sep 10 2011 [this is correct, see the g.f., Joerg Arndt, Sep 29 2013]
Also, A033954 and positive members of A001107 interleaved. - Omar E. Pol, Aug 04 2012
Also, numbers m such that 16*m+9 is a square. After 1, therefore, there are no squares in this sequence. - Bruno Berselli, Jan 07 2016
Convolution of the sequences A047522 and A059841. - Ilya Gutkovskiy, Mar 16 2017
Numbers k such that the concatenation k5625 is a square. - Bruno Berselli, Nov 07 2018
Exponents in expansion of Product_{n >= 1} (1 + x^(8*n-7))*(1 + x^(8*n-1))*(1 - x^(8*n)) = 1 + x + x^7 + x^10 + x^22 + .... - Peter Bala, Dec 10 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Neville Holmes, More Gemometric Integer Sequences. [Wayback Machine copy]
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A011848, A014493, A074378, A118277, A165998.
Cf. A001107 (10-gonal numbers).
Column 6 of A195152.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), this sequence (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]

[n^2+n-1/4+(-1)^n/4+n*(-1)^n/2: n in [0..50]]; // Vincenzo Librandi, Sep 29 2013

CoefficientList[Series[x(1 +6x +x^2)/((1-x)(1-x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 29 2013 *)
LinearRecurrence[{1,2,-2,-1,1}, {0,1,7,10,22}, 50] (* G. C. Greubel, Nov 07 2018 *)

PARI
```
a(n)=(2*n+3-4*(n%2))*(n-n\2)
```

concat([0],Vec(x*(1 + 6*x + x^2)/((1 - x)*(1 - x^2)^2) +O(x^50))) \\ Indranil Ghosh, Mar 16 2017

def A074377(n): return (n+1>>1)*((n<<1)+(-1 if n&1 else 3)) # Chai Wah Wu, Mar 11 2025

(n(n+1)-2)/4 where n(n+1)/2 is odd.
G.f.: x*(1+6*x+x^2)/((1-x)*(1-x^2)^2). - Michael Somos, Mar 04 2003
a(2*k) = k*(4*k+3); a(2*k+1) = (2*k+1)^2+k. - Benoit Jubin, Feb 05 2009
a(n) = n^2+n-1/4+(-1)^n/4+n*(-1)^n/2. - R. J. Mathar, Oct 08 2011
Sum_{n>=1} 1/a(n) = (4 + 3*Pi)/9. - Vaclav Kotesovec, Oct 05 2016
E.g.f.: exp(x)*x^2 + (2*exp(x) - exp(-x)/2)*x - sinh(x)/2. - Ilya Gutkovskiy, Mar 16 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 4/9. - Amiram Eldar, Feb 28 2022
a(n) = (n+1)*(2*n-1)/2 if n is odd and a(n) = n*(2*n+3)/2 if n is even. - Chai Wah Wu, Mar 11 2025

New name from T. D. Noe, Apr 21 2006

Formula in sequence name from Omar E. Pol, May 28 2012

A195161 Multiples of 8 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 8, 3, 16, 5, 24, 7, 32, 9, 40, 11, 48, 13, 56, 15, 64, 17, 72, 19, 80, 21, 88, 23, 96, 25, 104, 27, 112, 29, 120, 31, 128, 33, 136, 35, 144, 37, 152, 39, 160, 41, 168, 43, 176, 45, 184, 47, 192, 49, 200, 51, 208, 53, 216, 55, 224, 57, 232, 59

Offset: 0

Omar E. Pol, Sep 10 2011

A008590 and A005408 interleaved. This is 8*n if n is even, n if n is odd, if n>=0.
Partial sums give the generalized 12-gonal (or dodecagonal) numbers A195162.
The moment generating function of p(x, m=2, n=1, mu=2) = 4*x*E(x, 2, 1), see A163931 and A274181, is given by M(a) = (- 4*log(1-a) - 4 * polylog(2, a))/a^2. The series expansion of M(a) leads to the sequence given above. - Johannes W. Meijer, Jul 03 2016
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 12-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Column 8 of A195151.
Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, this sequence, A195312.
Cf. A144433.

&cat[[8*n, 2*n+1]: n in [0..30]]; // Vincenzo Librandi, Sep 27 2011

a := proc(n): (6*(-1)^n+10)*n/4 end: seq(a(n), n=0..59); # Johannes W. Meijer, Jul 03 2016

With[{nn=30},Riffle[8*Range[0,nn],2*Range[0,nn]+1]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,8,3},60] (* Harvey P. Dale, Nov 24 2013 *)

concat(0, Vec(x*(1+8*x+x^2)/((1-x)^2*(1+x)^2) + O(x^99))) \\ Altug Alkan, Jul 04 2016

a(2n) = 8n, a(2n+1) = 2n+1. [corrected by Omar E. Pol, Jul 26 2018]
a(n) = (6*(-1)^n+10)*n/4. - Vincenzo Librandi, Sep 27 2011
a(n) = 2*a(n-2)-a(n-4). G.f.: x*(1+8*x+x^2)/((1-x)^2*(1+x)^2). - Colin Barker, Aug 11 2012
From Ilya Gutkovskiy, Jul 03 2016: (Start)
a(m*2^k) = m*2^(k+2), k>0.
E.g.f.: x*(4*sinh(x) + cosh(x)).
Dirichlet g.f.: 2^(-s)*(2^s + 6)*zeta(s-1). (End)
Multiplicative with a(2^e) = 4*2^e, a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
a(n) = A144433(n-1) for n > 1. - Georg Fischer, Oct 14 2018

A195140 Multiples of 5 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 5, 3, 10, 5, 15, 7, 20, 9, 25, 11, 30, 13, 35, 15, 40, 17, 45, 19, 50, 21, 55, 23, 60, 25, 65, 27, 70, 29, 75, 31, 80, 33, 85, 35, 90, 37, 95, 39, 100, 41, 105, 43, 110, 45, 115, 47, 120, 49, 125, 51, 130, 53, 135, 55, 140, 57, 145, 59, 150, 61, 155, 63

Offset: 0

Omar E. Pol, Sep 10 2011

This is 5*n/2 if n is even, n if n is odd.
Partial sums give the generalized enneagonal numbers A118277.
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized enneagonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

A008587 and A005408 interleaved.
Column 5 of A195151.
Cf. Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, this sequence, zero together with A165998, A195159, A195161, A195312.
Cf. A047336, A118277.

&cat[[5*n,2*n+1]: n in [0..31]]; // Bruno Berselli, Sep 27 2011

With[{nn=40},Riffle[5*Range[0,nn],Range[1,2nn+1,2]]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,1,5,3},80] (* Harvey P. Dale, Dec 15 2014 *)

a(n)=(7+3*(-1)^n)*n/4 \\ Charles R Greathouse IV, Oct 07 2015

a(2n) = 5n, a(2n+1) = 2n+1.
G.f.: x*(1+5*x+x^2) / ((x-1)^2*(x+1)^2). - Alois P. Heinz, Sep 26 2011
From Bruno Berselli, Sep 27 2011: (Start)
a(n) = (7+3*(-1)^n)*n/4.
a(n) = -a(-n) = a(n-2)*n/(n-2) = 2*a(n-2)-a(n-4).
a(n) + a(n-1) = A047336(n). (End)
Multiplicative with a(2^e) = 5*2^(e-1), a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 3/2^s). - Amiram Eldar, Oct 25 2023

Corrected and edited by Alois P. Heinz, Sep 25 2011

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 3, 1, 1, 0, 0, 3, 2, 1, 0, 5, 2, 3, 3, 1, 0, 0, 5, 4, 3, 4, 1, 0, 7, 3, 5, 6, 3, 5, 1, 0, 0, 7, 6, 5, 8, 3, 6, 1, 0, 9, 4, 7, 9, 5, 10, 3, 7, 1, 0, 0, 9, 8, 7, 12, 5, 12, 3, 8, 1, 0, 11, 5, 9, 12, 7, 15, 5, 14, 3, 9, 1, 0, 0, 11, 10, 9, 16, 7

Offset: 0

Omar E. Pol, Sep 14 2011

Also square array T(n,k) read by antidiagonals in which column k lists the multiples of k and the odd numbers interleaved, n>=0, k>=0. Also square array T(n,k) read by antidiagonals in which if n is even then row n lists the multiples of (n/2), otherwise if n is odd then row n lists a constant sequence: the all n's sequence. Partial sums of the numbers of column k give the column k of A195152. Note that if k >= 1 then partial sums of the numbers of the column k give the generalized m-gonal numbers, where m = k + 4.

All columns are multiplicative. - Andrew Howroyd, Jul 23 2018

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,...
.  3,   3,   3,   3,   3,   3,   3,   3,   3,   3,...
.  0,   2,   4,   6,   8,  10,  12,  14,  16,  18,...
.  5,   5,   5,   5,   5,   5,   5,   5,   5,   5,...
.  0,   3,   6,   9,  12,  15,  18,  21,  24,  27,...
.  7,   7,   7,   7,   7,   7,   7,   7,   7,   7,...
.  0,   4,   8,  12,  16,  20,  24,  28,  32,  36,...
.  9,   9,   9,   9,   9,   9,   9,   9,   9,   9,...
.  0,   5,  10,  15,  20,  25,  30,  35,  40,  45,...
...

Rows: A000004, A000012, A001477, A010701, A005843, A010716, A008585, A010727, A008586, A010734, A008587.

Columns k: A026741 (k=1), A001477 (k=2), zero together with A080512 (k=3), A022998 (k=4), A195140 (k=5), zero together with A165998 (k=6), A195159 (k=7), A195161 (k=8), A195312 k=(9), A195817 (k=10), A317311 (k=11), A317312 (k=12), A317313 (k=13), A317314 k=(14), A317315 (k=15), A317316 (k=16), A317317 (k=17), A317318 (k=18), A317319 k=(19), A317320 (k=20), A317321 (k=21), A317322 (k=22), A317323 (k=23), A317324 k=(24), A317325 (k=25), A317326 (k=26).

T(n,k) = n*((k-2)*(-1)^n+k+2)/4 \\ Andrew Howroyd, Jul 23 2018

A195159 Multiples of 7 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 7, 3, 14, 5, 21, 7, 28, 9, 35, 11, 42, 13, 49, 15, 56, 17, 63, 19, 70, 21, 77, 23, 84, 25, 91, 27, 98, 29, 105, 31, 112, 33, 119, 35, 126, 37, 133, 39, 140, 41, 147, 43, 154, 45, 161, 47, 168, 49, 175, 51, 182, 53, 189, 55, 196, 57, 203, 59, 210, 61

Offset: 0

Omar E. Pol, Sep 10 2011

This is 7*n if n is even, n if n is odd, if n>=0.
Partial sums give the generalized 11-gonal (or hendecagonal) numbers A195160.
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 11-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008589 and A005408 interleaved.
Column k=7 of A195151.
Cf. Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, this sequence, A195161.
Cf. A056020, A195160.

&cat[[7*n, 2*n+1]: n in [0..40]]; // Vincenzo Librandi, Sep 27 2011

Table[If[EvenQ[n], 7(n/2), n], {n, 0, 61}] (* Alonso del Arte, Sep 14 2011 *)
With[{nn=40},Riffle[7*Range[0,nn],Range[1,2nn,2]]] (* Harvey P. Dale, Aug 01 2019 *)

a(n)=(5*(-1)^n+9)*n/4 \\ Charles R Greathouse IV, Oct 07 2015

a(2n) = 7n, a(2n+1) = 2n+1. [corrected by Omar E. Pol, Jul 26 2018]
From Bruno Berselli, Sep 14 2011: (Start)
G.f.: x*(1+7*x+x^2)/((1-x)^2*(1+x)^2).
a(n) = (5*(-1)^n+9)*n/4.
a(n) + a(n-1) = A056020(n). (End)
Multiplicative with a(2^e) = 7*2^(e-1), a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 5/2^s). - Amiram Eldar, Oct 25 2023

A195312 Multiples of 9 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 9, 3, 18, 5, 27, 7, 36, 9, 45, 11, 54, 13, 63, 15, 72, 17, 81, 19, 90, 21, 99, 23, 108, 25, 117, 27, 126, 29, 135, 31, 144, 33, 153, 35, 162, 37, 171, 39, 180, 41, 189, 43, 198, 45, 207, 47, 216, 49, 225, 51, 234, 53, 243, 55, 252, 57, 261, 59, 270, 61

Offset: 0

Omar E. Pol, Sep 14 2011

Partial sums give the generalized 13-gonal (or tridecagonal) numbers A195313.

a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized 13-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Column 9 of A195151.
Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, A195161, this sequence.
Cf. A195313, A175885.

/* By definition */ &cat[[9*n,2*n+1]: n in [0..33]]; // Bruno Berselli, Sep 16 2011

With[{nn=30},Riffle[9Range[0,nn],Range[1,2nn+1,2]]] (* Harvey P. Dale, Sep 24 2011 *)

a(n)=(7*(-1)^n+11)*n/4 \\ Charles R Greathouse IV, Oct 07 2015

From Bruno Berselli, Sep 15 2011:  (Start)
G.f.: x*(1+9*x+x^2)/((1-x)^2*(1+x)^2).
a(n) = (7*(-1)^n+11)*n/4.
a(n) + a(n-1) = A175885(n).
Sum_{i=0..n} a(i) = A195313(n). (End)
Multiplicative with a(2^e) = 9*2^(e-1), a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 7/2^s). - Amiram Eldar, Oct 25 2023
E.g.f.: x*(cosh(x) + 9*sinh(x)/2). - Stefano Spezia, Jun 12 2025

A195817 Multiples of 10 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 10, 3, 20, 5, 30, 7, 40, 9, 50, 11, 60, 13, 70, 15, 80, 17, 90, 19, 100, 21, 110, 23, 120, 25, 130, 27, 140, 29, 150, 31, 160, 33, 170, 35, 180, 37, 190, 39, 200, 41, 210, 43, 220, 45, 230, 47, 240, 49, 250, 51, 260, 53, 270, 55, 280, 57, 290, 59, 300

Offset: 0

Omar E. Pol, Sep 29 2011

A008592 and A005408 interleaved.
Partial sums give the generalized 14-gonal (or tetradecagonal) numbers A195818.
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized 14-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Column 10 of A195151.
Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, A195161, A195312, this sequence.
Cf. A091998, A195152.

[(2*(-1)^n+3)*n: n in [0..60]]; // Vincenzo Librandi, Sep 30 2011

With[{nn=30},Riffle[10*Range[0,nn],Range[1,2*nn+1,2]]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,10,3},70] (* Harvey P. Dale, Nov 24 2013 *)

a(n) = (2*(-1)^n+3)*n; \\ Andrew Howroyd, Jul 23 2018

a(n) = (2*(-1)^n+3)*n. - Vincenzo Librandi, Sep 30 2011
From Bruno Berselli, Sep 30 2011: (Start)
G.f.: x*(1+10*x+x^2)/((1-x)^2*(1+x)^2).
a(n) = -a(-n) = a(n-2)*n/(n-2) = 2*a(n-2)-a(n-4).
a(n) * a(n+1) = a(n(n+1)).
a(n) + a(n+1) = A091998(n+1). (End)
a(0)=0, a(1)=1, a(2)=10, a(3)=3, a(n)=2*a(n-2)-a(n-4). - Harvey P. Dale, Nov 24 2013
Multiplicative with a(2^e) = 5*2^e, a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 2^(3-s)). - Amiram Eldar, Oct 25 2023

A215466 Expansion of x(1-4x+x^2) / ( (x^2-7x+1)(x^2-3*x+1) ).

Original entry on oeis.org

0, 1, 6, 38, 252, 1705, 11628, 79547, 544824, 3733234, 25585230, 175356611, 1201893336, 8237850373, 56462937882, 387002396990, 2652553009008, 18180866487757, 124613506702404, 854113665498719, 5854182112700460

Offset: 0

R. J. Mathar, Aug 11 2012

From Peter Bala, Aug 05 2019: (Start)
Let  U(n;P,Q), where P and Q are integer parameters, denote the Lucas sequence of the first kind. Then, excluding the case P = -1, the sequence ( U(n;P,1) + U(2*n;P,1) )/(P + 1) is a fourth-order linear divisibility sequence with o.g.f. x*(1 - 2*(P - 1)*x + x^2)/((1 - P*x + x^2)*(1 - (P^2 - 2)*x + x^2)). This is the case P = 3. See A000027 (P = 2), A165998 (P = -2) and A238536 (P = -3).
More generally, the sequence U(n;P,1) + U(2*n;P,1) + ... + U(k*n;P,1) is a linear divisibility sequence of order 2*k. As an example, see A273625 (P = 3, k = 3 and then sequence normalized with initial term 1). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Lucas sequence
Peter Bala, Divisibility sequences from strong divisibility sequences
E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Odd and even linear divisibility sequences of order 4, INTEGERS, 2015, #A33.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (10,-23,10,-1).

Cf. A000032, A000045, A001906, A033888, A165998, A215465, A238536, A273625.

I:=[0,1,6,38]; [n le 4 select I[n] else 10*Self(n-1)-23*Self(n-2)+10*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 23 2012

/* By definition: */ m:=20; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1-4*x+x^2)/((x^2-7*x+1)*(x^2-3*x+1)))); // Bruno Berselli, Dec 24 2012

A215466 := proc(n)
    if type(n,'even') then
        A000032(n)*combinat[fibonacci](3*n)/4 ;
    else
        combinat[fibonacci](n)*A000032(3*n)/4 ;
    end if;
end proc:

CoefficientList[Series[x*(1 - 4*x + x^2)/((x^2 - 7*x + 1)*(x^2 - 3*x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 23 2012 *)
LinearRecurrence[{10,-23,10,-1},{0,1,6,38},30] (* Harvey P. Dale, Nov 02 2015 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,10,-23,10]^n*[0;1;6;38])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

{a(n) = my(w=quadgen(5)^(2*n)); imag(w^2+w)/4}; /* Michael Somos, Dec 29 2022 */

a(n) = L(n)*F(3n)/4 if n even, = F(n)*L(3n)/4 if n odd, where L=A000032, F=A000045.
a(n) = 3*A004187(n)/4 + A001906(n)/4.
a(n) = 10*a(n-1) - 23*a(n-2) + 10*a(n-3) - a(n-4), a(0)=0, a(1)=1, a(2)=6, a(3)=38. - Harvey P. Dale, Nov 02 2015
a(n) = (1/4)*(Fibonacci(2*n) + Fibonacci(4*n)) = (1/4)*(A001906(n) + A033888(n)). - Peter Bala, Aug 05 2019
E.g.f.: exp(5*x/2)*(cosh(x)+exp(x)*cosh(sqrt(5)*x))*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Aug 17 2019
a(n) = -a(-n) for all n in Z. - Michael Somos, Dec 29 2022

A317311 Multiples of 11 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 11, 3, 22, 5, 33, 7, 44, 9, 55, 11, 66, 13, 77, 15, 88, 17, 99, 19, 110, 21, 121, 23, 132, 25, 143, 27, 154, 29, 165, 31, 176, 33, 187, 35, 198, 37, 209, 39, 220, 41, 231, 43, 242, 45, 253, 47, 264, 49, 275, 51, 286, 53, 297, 55, 308, 57, 319, 59, 330, 61, 341, 63, 352, 65, 363, 67, 374, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 15-gonal numbers (A277082).

a(n) is also the length of the n-th line segment of the rectangular spiral wh0se vertices are the generalized 15-gonal numbers.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008593 and A005408 interleaved.
Column 11 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A277082.

{0}~Join~Riffle[2 Range@ # - 1, 11 Range@ #] &@ 35 (* or *)
CoefficientList[Series[x (1 + 11 x + x^2)/((1 - x)^2*(1 + x)^2), {x, 0, 69}], x] (* Michael De Vlieger, Jul 26 2018 *)
LinearRecurrence[{0,2,0,-1},{0,1,11,3},90] (* Harvey P. Dale, Aug 28 2022 *)

concat(0, Vec(x*(1 + 11*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jul 26 2018

a(2n) = 11*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 26 2018: (Start)
G.f.: x*(1 + 11*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 11*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 9/2^s). - Amiram Eldar, Oct 25 2023
a(n) = (13 + 9*(-1)^n)*n/4. - Aaron J Grech, Aug 20 2024

A317312 Multiples of 12 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 12, 3, 24, 5, 36, 7, 48, 9, 60, 11, 72, 13, 84, 15, 96, 17, 108, 19, 120, 21, 132, 23, 144, 25, 156, 27, 168, 29, 180, 31, 192, 33, 204, 35, 216, 37, 228, 39, 240, 41, 252, 43, 264, 45, 276, 47, 288, 49, 300, 51, 312, 53, 324, 55, 336, 57, 348, 59, 360, 61, 372, 63, 384, 65, 396, 67, 408, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 16-gonal numbers (A274978).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 16-gonal numbers.

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008594 and A005408 interleaved.
Column 12 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15).
Cf. A274978.

{0}~Join~Riffle[2 Range@ # - 1, 12 Range@ #] &@ 35 (* or *)
CoefficientList[Series[x (1 + 12 x + x^2)/((1 - x)^2*(1 + x)^2), {x, 0, 69}], x] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 12, 3}, 70] (* Michael De Vlieger, Jul 26 2018 *)

a(2n) = 12*n, a(2n+1) = 2*n + 1.
From Michael De Vlieger, Jul 26 2018: (Start)
G.f.: x*(1 + 12*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 3*2^(e+1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 5*2^(1-s)). - Amiram Eldar, Oct 25 2023
a(n) = (7 + 5*(-1)^n)*n/2. - Aaron J Grech, Aug 20 2024

cp's OEIS Frontend

A074377 Generalized 10-gonal numbers: m*(4*m - 3) for m = 0, +- 1, +- 2, +- 3, ...

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A195161 Multiples of 8 and odd numbers interleaved.

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A195140 Multiples of 5 and odd numbers interleaved.

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A195151 Square array read by antidiagonals upwards: T(n,k) = n*((k-2)*(-1)^n+k+2)/4, n >= 0, k >= 0.

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A195159 Multiples of 7 and odd numbers interleaved.

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A195312 Multiples of 9 and odd numbers interleaved.

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A195817 Multiples of 10 and odd numbers interleaved.

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A074377 Generalized 10-gonal numbers: m(4m - 3) for m = 0, +- 1, +- 2, +- 3, ...

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

A215466 Expansion of x(1-4x+x^2) / ( (x^2-7x+1)(x^2-3*x+1) ).