A077221 -id:A077221 - cp's OEIS frontend

A139595 A139277(n) followed by A139273(n+1).

0, 5, 13, 26, 42, 63, 87, 116, 148, 185, 225, 270, 318, 371, 427, 488, 552, 621, 693, 770, 850, 935, 1023, 1116, 1212, 1313, 1417, 1526, 1638, 1755, 1875, 2000, 2128, 2261, 2397, 2538, 2682, 2831, 2983, 3140, 3300, 3465, 3633, 3806

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 5,... and the same line from 0, in the direction 0, 13,..., in the square spiral whose vertices are the triangular numbers A000217.

			Array begins:
0, 5
13, 26
42, 63
87, 116

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1)

Cf. A000217, A046092, A139273, A139277, A077221, A139591, A139592, A139593, A139596, A139597, A139598.

LinearRecurrence[{2,0,-2,1},{0,5,13,26},50] (* Harvey P. Dale, Jul 31 2021 *)

Array read by rows: row n gives 8*n^2 + 5n, 8*(n+1)^2 - 3(n+1).

G.f.: -x*(5+3*x) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Feb 13 2011

A139597 A139278(n) followed by A139274(n+1).

Original entry on oeis.org

0, 7, 15, 30, 46, 69, 93, 124, 156, 195, 235, 282, 330, 385, 441, 504, 568, 639, 711, 790, 870, 957, 1045, 1140, 1236, 1339, 1443, 1554, 1666, 1785, 1905, 2032, 2160, 2295, 2431, 2574, 2718, 2869, 3021, 3180, 3340, 3507, 3675, 3850

Offset: 0

Omar E. Pol, May 03 2008

Sequence found by reading the line from 0, in the direction 0, 7,... and the line from 15, in the direction 15, 46,..., in the square spiral whose vertices are the triangular numbers A000217.

			Array begins:
0, 7
15, 30
46, 69
93, 124

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A046092, A139274, A139278, A077221, A139591, A139592, A139593, A139595, A139596, A139598.

Array read by rows: row n gives 8*n^2 + 7n, 8*(n+1)^2 - (n+1).

a(n) = (3-3*(-1)^n+14*n+8*n^2)/4. a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). G.f.: x*(7+x)/((1-x)^3*(1+x)). [Colin Barker, Jul 22 2012]

A244082 a(n) = 32*n^2.

Original entry on oeis.org

0, 32, 128, 288, 512, 800, 1152, 1568, 2048, 2592, 3200, 3872, 4608, 5408, 6272, 7200, 8192, 9248, 10368, 11552, 12800, 14112, 15488, 16928, 18432, 20000, 21632, 23328, 25088, 26912, 28800, 30752, 32768, 34848, 36992, 39200, 41472, 43808, 46208, 48672, 51200

Offset: 0

Wesley Ivan Hurt, Jun 19 2014

Geometric connections of a(n) to the area and perimeter of a square.
Area:
. half the area of a square with side 8n (cf. A008590);
. area of a square with diagonal 8n (cf. A008590);
. twice the area of a square with side 4n (cf. A008586);
. four times the area of a square with diagonal 4n (cf. A008586);
. eight times the area of a square with side 2n (cf. A005843);
. sixteen times the area of a square with diagonal 2n (cf. A005843);
. thirty two times the area of a square with side n (cf. A001477);
. sixty four times the area of a square with diagonal n (cf. A001477).
Perimeter:
. perimeter of a square with side 8n^2 (cf. A139098);
. twice the perimeter of a square with side 4n^2 (cf. A016742);
. four times the perimeter of a square with side 2n^2 (cf. A001105);
. eight times the perimeter of a square with side n^2 (cf. A000290).
Sequence found by reading the line from 0, in the direction 0, 32, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - Omar E. Pol, May 10 2018

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Pentasection of A077221, A181900.
Cf. A000290, A001105, A010021, A016742, A016802, A139098.
Cf. A001477, A005843, A008586, A008590, A139098, A174312.

Magma
```
[32*n^2 : n in [0..50]];
```

A244082:=n->32*n^2; seq(A244082(n), n=0..50);

32 Range[0, 50]^2 (* or *)
Table[32 n^2, {n, 0, 50}] (* or *)
CoefficientList[Series[32 x (1 + x)/(1 - x)^3, {x, 0, 30}], x]

a(n)=32*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 32*x*(1+x)/(1-x)^3.
a(n) =  2 * A016802(n).
a(n) =  4 * A139098(n).
a(n) =  8 * A016742(n).
a(n) = 16 * A001105(n).
a(n) = 32 * A000290(n).
a(n) = A010021(n) - 2 for n > 0. - Bruno Berselli, Jun 24 2014
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Nov 19 2021
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 32*x*(1 + x)*exp(x).
a(n) = n*A174312(n) = A139098(2*n). (End)

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.

Original entry on oeis.org

1, 2, 3, 23, 59, 111, 179, 263, 363, 479, 611, 759, 923, 1103, 1299, 1511, 1739, 1983, 2243, 2519, 2811, 3119, 3443, 3783, 4139, 4511, 4899, 5303, 5723, 6159, 6611, 7079, 7563, 8063, 8579, 9111, 9659, 10223, 10803, 11399, 12011, 12639, 13283, 13943

Offset: 0

Bruno Berselli, Sep 13 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the line 1, 2, 3, 23,.. in the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

This is a subsequence of A110326 (without signs) and A047838 (apart from the second term, 2).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195241.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217.

Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195605 [incomplete list].

m:=44; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+19*x^3-3*x^4)/(1-x)^3));

CoefficientList[Series[(1 - x + 19 x^3 - 3 x^4)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{3,-3,1},{1,2,3,23,59},50] (* Harvey P. Dale, Dec 04 2022 *)

makelist(coeff(taylor((1-x+19*x^3-3*x^4)/(1-x)^3, x, 0, n), x, n), n, 0, 43);

Vec((1-x+19*x^3-3*x^4)/(1-x)^3+O(x^44))

G.f.: (1-x+19*x^3-3*x^4)/(1-x)^3.

a(n) = 8*n^2-20*n+11 for n>1; a(0)=1, a(1)=2.

A275496 a(n) = n^2(2n^2 + (-1)^n).

Original entry on oeis.org

0, 1, 36, 153, 528, 1225, 2628, 4753, 8256, 13041, 20100, 29161, 41616, 56953, 77028, 101025, 131328, 166753, 210276, 260281, 320400, 388521, 468996, 559153, 664128, 780625, 914628, 1062153, 1230096, 1413721, 1620900, 1846081, 2098176, 2370753, 2673828

Offset: 0

Daniel Poveda Parrilla, Jul 30 2016

All terms of this sequence are triangular numbers. Graphically, for each term of the sequence, one corner of the square of squares (4th power) will be part of the corresponding triangle's hypotenuse if the term is an odd number. Otherwise, it will not be part of it.
a(A000129(n)) is a square triangular number.
a(2^((A000043(n) - 1)/2)) - 2^A000043(n) is a perfect number.

			a(5) = 5^4 + Sum_{k=0..(5^2 - (5 mod 2))} 2k = 625 + Sum_{k=0..(25 - 1)} 2k = 625 + 600 = 1225.
a(12) = 12^4 + Sum_{k=0..(12^2 - (12 mod 2))} 2k = 20736 + Sum_{k=0..(144 - 0)} 2k = 20736 + 20880 = 41616.

Colin Barker and Daniel Poveda Parrilla, Table of n, a(n) for n = 0..46340 [n = 1 through 1000 by Colin Barker, Aug 02 2016; and n=1001 to 46340 by Daniel Poveda Parrilla, Aug 04 2016]
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A000040, A000129, A000217, A001110, A077221, A093954, A275543.

Table[n^2 ((-1)^n + 2 n^2), {n, 0, 34}] (* or *)
CoefficientList[Series[x (1 + 34 x + 79 x^2 + 156 x^3 + 79 x^4 + 34 x^5 +
x^6)/((1 - x)^5 (1 + x)^3), {x, 0, 34}], x] (* Michael De Vlieger, Aug 01 2016 *)
LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{0,1,36,153,528,1225,2628,4753},40] (* Harvey P. Dale, Sep 10 2016 *)

a(n)=n=n^2; if(n%2,2*n-1,2*n+1)*n \\ Charles R Greathouse IV, Jul 30 2016

concat(0, Vec(x*(1+34*x+79*x^2+156*x^3+79*x^4+34*x^5+x^6)/((1-x)^5*(1+x)^3) + O(x^100))) \\ Colin Barker, Aug 01 2016

a(n) = n^4 + Sum_{k=0..(n^2 - (n mod 2))} 2k.
a(n) = A275543(n)*(n^2).
From Colin Barker, Aug 01 2016 and Aug 04 2016: (Start)
a(n) = n^2*(2*n^2 + (-1)^n).
a(n) = 2*n^4 + n^2 for n even.
a(n) = 2*n^4 - n^2 for n odd.
G.f.: x*(1 +34*x +79*x^2 +156*x^3 +79*x^4 +34*x^5 +x^6) / ((1-x)^5*(1+x)^3).
(End)
a(n) = n^2*A000217(2n-1) + 2n*A000217(n-(n mod 2)) for n > 0.
E.g.f.: x*(2*(1 + 7*x + 6*x^2 + x^3)*exp(x) - exp(-x)). - G. C. Greubel, Aug 05 2016
a(n) = A000217(A077221(n)).
a(n) = (A001844(A077221(n)) - 1)/4.
Sum_{n>=1} 1/a(n) = 1 - Pi^2/12 + (tan(c) - coth(c))*c, where c = Pi/(2*sqrt(2)) is A093954. - Amiram Eldar, Aug 21 2022

New name from Colin Barker, Aug 04 2016

A275543 A081585 and A069129 interleaved.

Original entry on oeis.org

1, 1, 9, 17, 33, 49, 73, 97, 129, 161, 201, 241, 289, 337, 393, 449, 513, 577, 649, 721, 801, 881, 969, 1057, 1153, 1249, 1353, 1457, 1569, 1681, 1801, 1921, 2049, 2177, 2313, 2449, 2593, 2737, 2889, 3041, 3201, 3361, 3529, 3697, 3873, 4049, 4233, 4417, 4609

Offset: 0

Daniel Poveda Parrilla, Aug 01 2016

a(A000129(n)) is a square.
(n^2)*a(n) = A275496(n) which is a triangular number.
(A000129(n)^2)*a(A000129(n)) = A275496(A000129(n)) = A001110(n) which is a square triangular number.
a(2n+1)/a(2n) is convergent to 1.

			a(1) = A275496(1) = 1.
a(5) = A275496(5)/25 = 1225/25 = 49.
a(7) = A275496(7)/49 = 4753/49 = 97.
a(12) = A275496(12)/144 = 41616/144 = 289.

Daniel Poveda Parrilla, Table of n, a(n) for n = 0..500000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081585(n) = a(2n), A069129(n) = a(2n + 1).

Cf. A000129, A000217, A000290, A001110, A077221, A093954, A275496.

CoefficientList[Series[(1 - x + 7 x^2 + x^3)/((1 - x)^3 (1 + x)), {x, 0, 48}], x] (* or as defined *)
Riffle[LinearRecurrence[{3, -3, 1}, {1, 9, 33}, #], FoldList[#1 + #2 &, 1, 16 Range@ #]] &@ 25 (* Michael De Vlieger, Aug 01 2016, after Vincenzo Librandi at A081585 and Robert G. Wilson v at A069129 *)

a(n)=(-1)^n + 2*n^2 \\ Charles R Greathouse IV, Aug 03 2016

Vec((1-x+7*x^2+x^3)/((1-x)^3*(1+x)) + O(x^100)) \\ Colin Barker, Aug 21 2016

a(0) = 1; a(n) = A275496(n)/(n^2) for n > 0.
From Colin Barker, Aug 01 2016: (Start)
a(n) = (2*n^2 + (-1)^n).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.
G.f.: (1 -x +7*x^2 +x^3) / ((1 - x)^3*(1 + x)).
(End)
From Daniel Poveda Parrilla, Aug 18 2016: (Start)
a(2n) = A077221(2n) + 1.
a(2n + 1) = A077221(2n + 1). (End)
Sum_{n>=0} 1/a(n) = (1 + (tan(c) + coth(c))*c)/2, where c = Pi/(2*sqrt(2)) is A093954. - Amiram Eldar, Aug 21 2022

A077222 a(1) = 1 and then alternately the smallest even and odd numbers not occurring earlier such that the sum of two successive terms is a square.

Original entry on oeis.org

1, 8, 17, 32, 49, 72, 9, 16, 33, 48, 73, 96, 25, 24, 57, 64, 105, 120, 169, 56, 65, 104, 121, 168, 193, 248, 41, 40, 81, 88, 137, 152, 209, 80, 89, 136, 153, 208, 233, 128, 97, 192, 249, 112, 113, 176, 185, 256, 273, 352, 177, 184, 257, 272, 353, 376, 465, 160

Offset: 0

Amarnath Murthy, Nov 03 2002

nonn

Cf. A077221.

Corrected and extended by John W. Layman, Jan 12 2005

A131925 2*A002024 - A000012(signed).

Original entry on oeis.org

1, 5, 3, 5, 7, 5, 9, 7, 9, 7, 9, 11, 9, 11, 9, 13, 11, 13, 11, 13, 11, 13, 15, 13, 15, 13, 15, 13, 17, 15, 17, 15, 17, 15, 17, 15, 17, 19, 17, 19, 17, 19, 17, 19, 17

Offset: 0

Gary W. Adamson, Jul 29 2007

Row sums = A077221: (1, 8, 17, 32, 49, 72, 97, ...).

			First few rows of the triangle:
   1;
   5,  3;
   5,  7,  5;
   9,  7,  9,  7;
   9, 11,  9, 11,  9;
  13, 11, 13, 11, 13, 11;
  13, 15, 13, 15, 13, 15, 13;
  ....

Cf. A002024, A000012, A077221.

2*A002024 - A000012 (signed + - + -, ... by columns).

A002024 = (1; 2,2; 3,3,3; ...); A000012(signed) = (1; -1,1; 1,-1,1; ...).

A270693 Alternating sum of centered 25-gonal numbers.

Original entry on oeis.org

1, -25, 51, -100, 151, -225, 301, -400, 501, -625, 751, -900, 1051, -1225, 1401, -1600, 1801, -2025, 2251, -2500, 2751, -3025, 3301, -3600, 3901, -4225, 4551, -4900, 5251, -5625, 6001, -6400, 6801, -7225, 7651, -8100, 8551, -9025, 9501, -10000, 10501

Offset: 0

Ilya Gutkovskiy, Mar 21 2016

The absolute value alternating sum of centered k-gonal numbers gives concentric k-gonal numbers.

More generally, the ordinary generating function for the alternating sum of centered k-gonal numbers is (1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^3).

OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A262221 (centered 25-gonal numbers).

Cf. A032527, A032528, A077043, A077221, A195041, A195042, A195045, A195046, A195047, A195048, A195049, A195058, A195142, A195043, A195143, A195145, A195146, A195147, A195148, A195149, A195158.

[((-1)^n*(50*n^2 + 100*n + 29) - 21)/8 : n in [0..40]]; // Wesley Ivan Hurt, Mar 21 2016

A270693:=n->((-1)^n*(50*n^2 + 100*n + 29) - 21)/8: seq(A270693(n), n=0..100); # Wesley Ivan Hurt, Sep 18 2017

LinearRecurrence[{-2, 0, 2, 1}, {1, -25, 51, -100}, 41]
Table[((-1)^n (50 n^2 + 100 n + 29) - 21)/8, {n, 0, 40}]

x='x+O('x^100); Vec((1-23*x+x^2)/((1-x)*(1+x)^3)) \\ Altug Alkan, Mar 21 2016

G.f.: (1 - 23*x + x^2)/((1 - x)*(1 + x)^3).
E.g.f.: (1/8)*(-21*exp(x) + (29 - 150*x + 50*x^2)*exp(-x)).
a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4).
a(n) = ((-1)^n*(50*n^2 + 100*n + 29) - 21)/8.

A237587 Triangle read by rows in which row n lists the first n odd squares in decreasing order.

Original entry on oeis.org

1, 9, 1, 25, 9, 1, 49, 25, 9, 1, 81, 49, 25, 9, 1, 121, 81, 49, 25, 9, 1, 169, 121, 81, 49, 25, 9, 1, 225, 169, 121, 81, 49, 25, 9, 1, 289, 225, 169, 121, 81, 49, 25, 9, 1, 361, 289, 225, 169, 121, 81, 49, 25, 9, 1, 441, 361, 289, 225, 169, 121, 81, 49, 25, 9, 1

Offset: 1

Omar E. Pol, Feb 16 2014

The sum of row n gives A000447(n).

The alternatig sum of row n gives A077221(n).

			Triangle begins:
1;
9,     1;
25,    9,   1;
49,   25,   9,   1;
81,   49,  25,   9,   1;
121,  81,  49,  25,   9,   1;
169, 121,  81,  49,  25,   9,   1;
225, 169, 121,  81,  49,  25,   9,   1;
289, 225, 169, 121,  81,  49,  25,   9,   1;
361, 289, 225, 169, 121,  81,  49,  25,   9,   1;
...

Cf. A000290, A000447, A016754, A055461, A077221.

T(n,k) = (2*(n-k+1)-1)^2, n >=1, 1<=k<=n.

cp's OEIS Frontend

A139595 A139277(n) followed by A139273(n+1).

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A139597 A139278(n) followed by A139274(n+1).

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A244082 a(n) = 32*n^2.

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A195241 Expansion of (1-x+19*x^3-3*x^4)/(1-x)^3.

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A275496 a(n) = n^2*(2*n^2 + (-1)^n).

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A275543 A081585 and A069129 interleaved.

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PARI

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A077222 a(1) = 1 and then alternately the smallest even and odd numbers not occurring earlier such that the sum of two successive terms is a square.

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A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.

A275496 a(n) = n^2(2n^2 + (-1)^n).