cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A017534 a(n) = (12*n + 1)^2.

Original entry on oeis.org

1, 169, 625, 1369, 2401, 3721, 5329, 7225, 9409, 11881, 14641, 17689, 21025, 24649, 28561, 32761, 37249, 42025, 47089, 52441, 58081, 64009, 70225, 76729, 83521, 90601, 97969, 105625, 113569, 121801
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m=3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), this sequence (m=12), A134934 (m=14).
Cf. A082043.

Programs

  • Magma
    I:=[1, 169, 625]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jul 07 2012
  • Mathematica
    CoefficientList[Series[(1+166*x+121*x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 07 2012 *)
LinearRecurrence[{3,-3,1},{1,169,625},30] (* Harvey P. Dale, Feb 27 2023 *)
  • PARI
    a(n)=(12*n+1)^2 \\ Charles R Greathouse IV, Jun 17 2017
  • SageMath
    [(12*n+1)^2 for n in range(51)] # G. C. Greubel, Dec 24 2022

Formula

G.f.: (1 + 166*x + 121*x^2 )/(1-x)^3. - R. J. Mathar, Mar 10 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 07 2012
E.g.f.: (1 + 168*x + 144*x^2)*exp(x). - G. C. Greubel, Dec 24 2022

A069131 Centered 18-gonal numbers.

Original entry on oeis.org

1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831, 8371, 8929, 9505, 10099, 10711, 11341, 11989, 12655, 13339, 14041, 14761, 15499, 16255, 17029, 17821
Offset: 1

Views

Author

Terrel Trotter, Jr., Apr 07 2002

Keywords

Comments

Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010
Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
From Lamine Ngom, Aug 19 2021: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section).
a(n) is a bisection of A195042.
a(n) is a trisection of A028387.
a(n) + 1 is promic (A002378).
a(n) + 2 is a trisection of A002061.
a(n) + 9 is the arithmetic mean of its neighbors.
4*a(n) + 5 is a square: A016945(n)^2. (End)

Examples

			a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.

Links

Crossrefs

Cf. centered polygonal numbers listed in A069190.
Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777,A016778, A016790, A010008, A008600, A002061.
Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715.

Programs

Formula

a(n) = 9*n^2 - 9*n + 1.
a(n) = 18*n + a(n-1) - 18 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - Harvey P. Dale, Jan 20 2014
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)).
Sum_{n>=1} a(n)/n! = 10*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End)
From Lamine Ngom, Aug 19 2021: (Start)
a(n) = 18*A000217(n) + 1 = 9*A002378(n) + 1.
a(n) = 3*A003215(n) - 2.
a(n) = A247792(n) - 9*n.
a(n) = A082040(n) + A304163(n) - a(n-1) = A016778(n) + A016790(n) - a(n-1), n > 0.
a(n) + a(n+1) = 2*A247792(n) = A010008(n), n > 0.
a(n+1) - a(n) = 18*n = A008600(n). (End)
From Leo Tavares, Oct 31 2021: (Start)
a(n)= A000290(n) + A139278(n-1)
a(n) = A069129(n) + A002378(n-1)
a(n) = A062786(n) + 8*A000217(n-1)
a(n) = A062786(n) + A033996(n-1)
a(n) = A060544(n) + 9*A000217(n-1)
a(n) = A060544(n) + A027468(n-1)
a(n) = A016754(n-1) + 10*A000217(n-1)
a(n) = A016754(n-1) + A124080
a(n) = A069099(n) + 11*A000217(n-1)
a(n) = A069099(n) + A152740(n-1)
a(n) = A003215(n-1) + 12*A000217(n-1)
a(n) = A003215(n-1) + A049598(n-1)
a(n) = A005891(n-1) + 13*A000217(n-1)
a(n) = A005891(n-1) + A152741(n-1)
a(n) = A001844(n) + 14*A000217(n-1)
a(n) = A001844(n) + A163756(n-1)
a(n) = A005448(n) + 15*A000217(n-1)
a(n) = A005448(n) + A194715(n-1). (End)
E.g.f.: exp(x)*(1 + 9*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023

A082043 Square array, A(n, k) = (k*n)^2 + 2*k*n + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 25, 16, 1, 1, 25, 49, 49, 25, 1, 1, 36, 81, 100, 81, 36, 1, 1, 49, 121, 169, 169, 121, 49, 1, 1, 64, 169, 256, 289, 256, 169, 64, 1, 1, 81, 225, 361, 441, 441, 361, 225, 81, 1, 1, 100, 289, 484, 625, 676, 625, 484, 289, 100, 1
Offset: 0

Views

Author

Paul Barry, Apr 03 2003

Keywords

Examples

			Array, A(n, k), begins as:
  1,   1,   1,    1,    1,    1,    1,    1,     1, ... A000012;
  1,   4,   9,   16,   25,   36,   49,   64,    81, ... A000290;
  1,   9,  25,   49,   81,  121,  169,  225,   289, ... A016754;
  1,  16,  49,  100,  169,  256,  361,  484,   625, ... A016778;
  1,  25,  81,  169,  289,  441,  625,  841,  1089, ... A016814;
  1,  36, 121,  256,  441,  676,  961, 1296,  1681, ... A016862;
  1,  49, 169,  361,  625,  961, 1369, 1849,  2401, ... A016922;
  1,  64, 225,  484,  841, 1296, 1849, 2500,  3249, ... A016994;
  1,  81, 289,  625, 1089, 1681, 2401, 3249,  4225, ... A017078;
  1, 100, 361,  784, 1369, 2116, 3025, 4096,  5329, ... A017174;
  1, 121, 441,  961, 1681, 2601, 3721, 5041,  6561, ... A017282;
  1, 144, 529, 1156, 2025, 3136, 4489, 6084,  7921, ... A017402;
  1, 169, 625, 1369, 2401, 3721, 5329, 7225,  9409, ... A017534;
  1, 196, 729, 1600, 2809, 4356, 6241, 8464, 11025, ... ;
Antidiagonals, T(n, k), begin as:
  1;
  1,   1;
  1,   4,   1;
  1,   9,   9,   1;
  1,  16,  25,  16,   1;
  1,  25,  49,  49,  25,   1;
  1,  36,  81, 100,  81,  36,   1;
  1,  49, 121, 169, 169, 121,  49,   1;
  1,  64, 169, 256, 289, 256, 169,  64,   1;
  1,  81, 225, 361, 441, 441, 361, 225,  81,   1;
  1, 100, 289, 484, 625, 676, 625, 484, 289, 100,  1;

Links

Crossrefs

Rows include A000290, A016754, A016778, A016814, A016862, A016922, A016994, A017078, A017174, A017282, A017402, A017534.
Diagonals include A000583, A058031, A062938, A082044 (main diagonal).
Diagonal sums (row sums if viewed as number triangle) are A082045.
Cf. A082039, A082045, A082046, A082105.

Programs

  • Magma
    A082043:= func< n,k | (k*(n-k))^2 +2*k*(n-k) +1 >;
[A082043(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 24 2022
  • Mathematica
    T[n_, k_]:= (k*(n-k))^2 +2*k*(n-k) +1;
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 24 2022 *)
  • SageMath
    def A082043(n,k): return (k*(n-k))^2 +2*k*(n-k) +1
flatten([[A082043(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Dec 24 2022

Formula

A(n, k) = (k*n)^2 + 2*k*n + 1 (square array).
T(n, k) = (k*(n-k))^2 + 2*k*(n-k) + 1 (number triangle).
A(k, n) = A(n, k).
T(n, n-k) = T(n, k).
A(n, n) = T(2*n, n) = A082044(n).
A(n, n-1) = T(2*n+1, n-1) = A058031(n), n >= 1.
A(n, n-2) = T(2*(n-1), n) = A000583(n-1), n >= 2.
A(n, n-3) = T(2*n-3, n) = A062938(n-3), n >= 3.
Sum_{k=0..n} T(n, k) = A082045(n) (diagonal sums).
Sum_{k=0..n} (-1)^k * T(n, k) = (1/4)*(1+(-1)^n)*(2 - 3*n). - G. C. Greubel, Dec 24 2022

A134934 a(n) = (14*n+1)^2.

Original entry on oeis.org

1, 225, 841, 1849, 3249, 5041, 7225, 9801, 12769, 16129, 19881, 24025, 28561, 33489, 38809, 44521, 50625, 57121, 64009, 71289, 78961, 87025, 95481, 104329, 113569, 123201, 133225, 143641, 154449, 165649, 177241, 189225, 201601, 214369, 227529, 241081
Offset: 0

Views

Author

Hans Isdahl, Jan 26 2008

Keywords

Comments

Number of rats in population after n years, starting with one rat at year 0 (see A016754 for more details).

Links

Crossrefs

Sequences of the form (m*n+1)^2: A000012 (m=0), A000290 (m=1), A016754 (m=2), A016778 (m=3), A016814 (m=4), A016862 (m=5), A016922 (m=6), A016994 (m=7), A017078 (m=8), A017174 (m=9), A017282 (m=10), A017402 (m=11), A017534 (m=12), this sequence (m=14).
Cf. A016754.

Programs

Formula

O.g.f.: (1+222*x+169*x^2)/(1-x)^3 = 169/(1-x) - 560/(1-x)^2 + 392/(1-x)^3. - R. J. Mathar, Jan 31 2008
a(n) = A016754(7*n).
E.g.f.: (1 + 224*x + 196*x^2)*exp(x). - G. C. Greubel, Dec 24 2022

A221596 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 1 or less.

Original entry on oeis.org

0, 0, 4, 0, 7, 8, 0, 10, 17, 16, 0, 13, 26, 49, 32, 0, 16, 35, 100, 139, 64, 0, 19, 44, 169, 342, 393, 128, 0, 22, 53, 256, 651, 1210, 1113, 256, 0, 25, 62, 361, 1068, 2715, 4240, 3151, 512, 0, 28, 71, 484, 1593, 5082, 11011, 14898, 8921, 1024, 0, 31, 80, 625, 2226, 8475
Offset: 1

Views

Author

R. H. Hardin Jan 20 2013

Keywords

Comments

Table starts
....0......0.......0........0........0.........0.........0.........0..........0
....4......7......10.......13.......16........19........22........25.........28
....8.....17......26.......35.......44........53........62........71.........80
...16.....49.....100......169......256.......361.......484.......625........784
...32....139.....342......651.....1068......1593......2226......2967.......3816
...64....393....1210.....2715.....5082......8475.....13056.....18987......26430
..128...1113....4240....11011....22912.....41401.....67936....103975.....150976
..256...3151...14898....45099...105586....210101....374342....615965.....954572
..512...8921...52306...184063...482204...1047967...2006006...3504371....5714456
.1024..25257..183684...752155..2210256...5267759..10894988..20352239...35218688
.2048..71507..645006..3072247.10115926..26387005..58789204.116958723..213700742
.4096.202449.2264978.12550859.46327024.132384353.318224626.675761541.1307528098

Examples

			Some solutions for n=6 k=4
..0....2....3....3....2....2....1....1....2....4....4....2....0....2....4....1
..0....1....4....2....3....3....2....1....3....3....4....2....1....3....3....2
..2....4....4....2....1....0....2....3....4....0....4....1....0....2....4....2
..2....4....1....4....1....0....1....2....0....1....3....4....0....2....4....1
..0....1....1....3....0....2....2....3....1....4....3....3....1....1....0....0
..0....2....1....2....1....3....1....2....2....3....2....3....0....1....0....0

Links

Crossrefs

Column 3 is A221568
Row 2 is A016777
Row 3 is A017257(n-1)
Row 4 is A016778

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>2
k=2: a(n) = 2*a(n-1) +2*a(n-2) +a(n-3) for n>4
k=3: a(n) = 3*a(n-1) +2*a(n-2) -a(n-3) +a(n-4)
k=4: a(n) = 3*a(n-1) +4*a(n-2) +6*a(n-4) +4*a(n-5) +4*a(n-6)
k=5: a(n) = 4*a(n-1) +3*a(n-2) -6*a(n-3) +19*a(n-4) +5*a(n-5) +a(n-6)
k=6: a(n) = 4*a(n-1) +5*a(n-2) -7*a(n-3) +33*a(n-4) +17*a(n-5) +24*a(n-6) -5*a(n-7) +2*a(n-8)
k=7: a(n) = 5*a(n-1) +3*a(n-2) -16*a(n-3) +65*a(n-4) -14*a(n-5) +23*a(n-6) +2*a(n-7) +8*a(n-8)
Empirical for row n:
n=2: a(n) = 3*n + 1
n=3: a(n) = 9*n - 1
n=4: a(n) = 9*n^2 + 6*n + 1
n=5: a(n) = 54*n^2 - 69*n + 63 for n>2
n=6: a(n) = 27*n^3 + 108*n^2 - 252*n + 267 for n>3
n=7: a(n) = 243*n^3 - 351*n^2 + 237*n + 127 for n>2

A016782 a(n) = (3*n+1)^6.

Original entry on oeis.org

1, 4096, 117649, 1000000, 4826809, 16777216, 47045881, 113379904, 244140625, 481890304, 887503681, 1544804416, 2565726409, 4096000000, 6321363049, 9474296896, 13841287201, 19770609664, 27680640625, 38068692544, 51520374361, 68719476736, 90458382169, 117649000000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A001014, A016777, A016778, A016779, A016780, A016781.

Programs

  • Magma
    [(3*n+1)^6: n in [0..30]]; // Vincenzo Librandi, Sep 21 2011
  • Mathematica
    Table[(3n+1)^6,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,4096,117649,1000000,4826809,16777216,47045881},20] (* Harvey P. Dale, Sep 30 2016 *)

Formula

a(n) = A001014(A016777(n)). - Michel Marcus, Jun 15 2016
From Ilya Gutkovskiy, Jun 15 2016: (Start)
G.f.: (1 + 4089*x + 88998*x^2 + 262438*x^3 + 154113*x^4 + 15177*x^5 + 64*x^6)/(1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). (End)
Sum_{n>=0} 1/a(n) = PolyGamma(5, 1/3)/87480. - Amiram Eldar, Mar 29 2022

A016785 a(n) = (3*n + 1)^9.

Original entry on oeis.org

1, 262144, 40353607, 1000000000, 10604499373, 68719476736, 322687697779, 1207269217792, 3814697265625, 10578455953408, 26439622160671, 60716992766464, 129961739795077, 262144000000000, 502592611936843, 922190162669056, 1628413597910449, 2779905883635712
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016777, A016778, A016779, A016780, A016781, A016782, A016783, A016784.

Programs

  • Magma
    [(3*n+1)^9 : n in [0..20]]; // Vincenzo Librandi, Sep 28 2011
  • Maple
    A016785:=n->(3*n+1)^9; seq(A016785(k), k=0..100); # Wesley Ivan Hurt, Nov 05 2013
  • Mathematica
    Table[(3*n+1)^9, {n,0,100}] (* Wesley Ivan Hurt, Nov 05 2013 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,262144,40353607,1000000000,10604499373,68719476736,322687697779,1207269217792,3814697265625,10578455953408},100] (* Harvey P. Dale, Aug 17 2014 *)

Formula

From Amiram Eldar, Mar 29 2022: (Start)
a(n) = A016777(n)^9 = A016779(n)^3.
Sum_{n>=0} 1/a(n) = 1618*Pi^9/(55801305*sqrt(3)) + 9841*zeta(9)/3^9. (End)

A016784 a(n) = (3*n+1)^8.

Original entry on oeis.org

1, 65536, 5764801, 100000000, 815730721, 4294967296, 16983563041, 54875873536, 152587890625, 377801998336, 852891037441, 1785793904896, 3512479453921, 6553600000000, 11688200277601, 20047612231936, 33232930569601, 53459728531456, 83733937890625
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A001016, A016777, A016778, A016779, A016780, A016781, A016782, A016783.

Programs

Formula

a(n)= A001016(A016777(n)). - Michel Marcus, Jun 15 2016
G.f.: (1 + 65527*x + 5175013*x^2 + 50476003*x^3 + 117758659*x^4 + 77404933*x^5 + 13270807*x^6 + 388321*x^7 + 256*x^8)/(1 - x)^9. - Ilya Gutkovskiy, Jun 16 2016
Sum_{n>=0} 1/a(n) = PolyGamma(7, 1/3)/33067440. - Amiram Eldar, Mar 29 2022

A221762 Numbers m such that 11*m^2 + 5 is a square.

Original entry on oeis.org

1, 2, 22, 41, 439, 818, 8758, 16319, 174721, 325562, 3485662, 6494921, 69538519, 129572858, 1387284718, 2584962239, 27676155841, 51569671922, 552135832102, 1028808476201, 11015040486199, 20524599852098, 219748673891878, 409463188565759
Offset: 1

Views

Author

Bruno Berselli, Jan 24 2013

Keywords

Comments

Corresponding squares are: 16, 49, 5329, 18496, 2119936, 7360369, 843728209, 2929407376, ... (subsequence of A016778).
The Diophantine equation 11*x^2+k = y^2, for |k|<11, has integer solutions with the following k values:
k = -10, the nonnegative x values are in A198947;
k = -8, " 2*A075839;
k = -7, " A221763;
k = -2, " A075839;
k = 1, " A001084;
k = 4, " A075844;
k = 5, " this sequence;
k = 9, " 3*A001084.
Also, the Diophantine equation h*x^2+5 = y^2 has infinitely many integer solutions for h = 5, 11, 19, 20, 29, 31, 41, 44, 55, 59, ...
a(n+1)/a(n) tends alternately to (1+sqrt(11))^2/10 and (4+sqrt(11))^2/5.
a(n+2)/a(n) tends to A176395^2/2.

Links

Crossrefs

Cf. A001084, A075839, A075844, A198947, A198949, A221763.
Cf. A049629 (numbers m such that 20*m^2 + 5 is a square), A075796 (numbers m such that 5*m^2 + 5 is a square).

Programs

  • Magma
    m:=24; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+2*x^2+x^3)/(1-20*x^2+x^4)));
  • Magma
    I:=[1,2,22,41]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Aug 18 2013
  • Maple
    A221762:=proc(q)
local n;
for n from 1 to q do if type(sqrt(11*n^2+5), integer) then print(n);
fi; od; end:
A221762(1000); # Paolo P. Lava, Feb 19 2013
  • Mathematica
    LinearRecurrence[{0, 20, 0, -1}, {1, 2, 22, 41}, 24]
CoefficientList[Series[(1 + 2 x + 2 x^2 + x^3)/(1 - 20 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 18 2013 *)
  • Maxima
    makelist(expand(((-11*(-1)^n+4*sqrt(11))*(10+3*sqrt(11))^floor(n/2)-(11*(-1)^n+4*sqrt(11))*(10-3*sqrt(11))^floor(n/2))/22), n, 1, 24);

Formula

G.f.: x*(1+2*x+2*x^2+x^3)/(1-20*x^2+x^4).
a(n) = -a(1-n) = ((-11*(-1)^n+4*t)*(10+3*t)^floor(n/2)-(11*(-1)^n+4*t)*(10-3*t)^floor(n/2))/22, where t=sqrt(11).
a(n) = 20*a(n-2) - a(n-4) for n>4, a(1)=1, a(2)=2, a(3)=22, a(4)=41.
a(n)*a(n-3)-a(n-1)*a(n-2) = -(3/2)*(9-7*(-1)^n).
a(n+1) + a(n-1) = A198949(n), with a(0)=-1.
2*a(n-1) - a(n) = A001084(n/2-1) for even n.

A247792 a(n) = 9*n^2 + 1.

Original entry on oeis.org

1, 10, 37, 82, 145, 226, 325, 442, 577, 730, 901, 1090, 1297, 1522, 1765, 2026, 2305, 2602, 2917, 3250, 3601, 3970, 4357, 4762, 5185, 5626, 6085, 6562, 7057, 7570, 8101, 8650, 9217, 9802, 10405, 11026, 11665, 12322, 12997, 13690, 14401, 15130, 15877, 16642, 17425, 18226, 19045, 19882
Offset: 0

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Author

Karl V. Keller, Jr., Sep 23 2014

Keywords

Comments

The odd numbers of the form 9n^2 + 1 are listed in A158591 (36n^2 + 1).
The even numbers of the form 9n^2 + 1 are given by 36x^2 - 36x + 10, x > 0.
Every integer n>0 give three perfect squares and consecutives from 2^2. The formulas for each value of n are: a(n)-6n, a(n)-1 and a(n)+6n. - Miquel Cerda, Sep 19 2016
These squares are, for n>0, A000290(3*n-1), 3*n and (3n+1) and the sum of them is 3*a(n) - 1. - Miquel Cerda, Sep 26 2016

Examples

			a(1) = (2^2 + 4^2)/2 = 3^2 + 1 = 10, a(2) = (5^2 + 7^2)/2 = 6^2 + 1 = 37, a(3) = (8^2 + 10^2)/2 = 9^2 + 1 = 82. - _Miquel Cerda_, Jun 25 2016

Links

Crossrefs

Cf. A016766, A158591 (36n^2 + 1), A156226 (primes of the form 9n^2 + 1).
Cf. also A000290.

Programs

Formula

a(n) = (3n)^2 + 1 = 9n^2 + 1 = A016766(n) + 1.
G.f.: (1+7*x+10*x^2)/(1-x)^3. - Vincenzo Librandi, Sep 27 2014
a(n) = ((3n-1)^2 + (3n+1)^2)/2 = (A016790(n-1) + A016778(n))/2. - Miquel Cerda, Jun 25 2016
From Ilya Gutkovskiy, Jun 25 2016: (Start)
E.g.f.: (1 + 9*x + 9*x^2)*exp(x).
Dirichlet g.f.: 9*zeta(s-2) + zeta(s).
Sum_{n>=0} 1/a(n) = (3 + Pi*coth(Pi/3))/6. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Wesley Ivan Hurt, Jun 25 2016
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/3)*csch(Pi/3))/2. - Amiram Eldar, Jul 15 2020
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/3)*sinh(sqrt(2)*Pi/3).
Product_{n>=1} (1 - 1/a(n)) = (Pi/3)*csch(Pi/3). (End)
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