A260862
Base-12 representation of a(n) is the concatenation of the base-12 representations of 1, 2, ..., n, n-1, ..., 1.
Original entry on oeis.org
0, 1, 169, 24649, 3553225, 511709641, 73686731209, 10610895808969, 1527969074670025, 220027547690625481, 31683966878707771849, 4562491230669011577289, 7883984846509322664831433, 163482309777203435651765004745, 3389969175540090458609916107975113
Offset: 0
a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = (12+1)^2 = 12^2 + 2*12 + 1 = 121_12, concatenation of (1, 2, 1).
a(13) = 123456789ab101110ba987654321_12 is the concatenation of (1, 2, 3, ..., 9, a, b, 10, 11, 10, b, ..., 1), where "b, 10, 11" are the base-12 representations of 11, 12, 13.
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a(n,b=12)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))
A260863
Base-13 representation of a(n) is the concatenation of the base-13 representations of 1, 2, ..., n, n-1, ..., 1.
Original entry on oeis.org
0, 1, 196, 33489, 5664400, 957345481, 161792190756, 27342890695849, 4620948663553600, 780940325907974961, 131978915101424183716, 22304436652439380447009, 3769449794266138309731600, 8281481197999449959084458465, 236527384496061684935031509169004
Offset: 0
a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = (13+1)^2 = 13^2 + 2*13 + 1 = 121_13, concatenation of (1, 2, 1).
a(14) = 123456789abc101110cba987654321_13 is the concatenation of (1, 2, 3, ..., 9, a, b, c, 10, 11, 10, c, ..., 1), where "c, 10, 11" are the base-13 representations of 12, 13, 14.
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a(n,b=13)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))
A261135
Decimal value a(n) of the binary number b(n) obtained by starting from n, sequentially concatenating all binary numbers down to 1 and then sequentially concatenating all binary numbers from 2 up to n.
Original entry on oeis.org
1, 22, 475, 20188, 1472229, 112441134, 8415721847, 1234163177336, 336405959636873, 95454487901599898, 26891544907776231851, 7513814442828887530172, 2084725487959034609380301, 574954374994698424018451678, 157752074714160807772190133743, 86177704805459827544499089423856
Offset: 1
a(1) = binary_to_decimal(1) = 1;
a(2) = binary_to_decimal(10110) = 22;
a(3) = binary_to_decimal(111011011) = 475;
a(4) = binary_to_decimal(100111011011100) = 20188.
Cf.
A173427 for an inverted ordering of repeated binary numbers.
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Table[d = IntegerDigits[#, 2] & /@ Range@ n; FromDigits[#, 2] &@
Flatten[{Flatten@ Reverse@ d, Flatten@ Rest@ d}, 1], {n, 16}] (* Michael De Vlieger, Aug 20 2015 *)
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